Quantum interest in two dimensions

TL;DR

This paper proves the quantum interest conjecture in two-dimensional Minkowski space by analyzing negative and positive energy pulses for a massless scalar field, providing explicit bounds on their timing and magnitude.

Contribution

It offers a rigorous proof of the quantum interest conjecture in two dimensions and derives explicit expressions for energy pulse constraints.

Findings

Explicit bounds on maximum time delay between pulses

Quantitative relation between over-compensation and separation

Proof valid for general energy distributions

Abstract

The quantum interest conjecture of Ford and Roman asserts that any negative-energy pulse must necessarily be followed by an over-compensating positive-energy one within a certain maximum time delay. Furthermore, the minimum amount of over-compensation increases with the separation between the pulses. In this paper, we first study the case of a negative-energy square pulse followed by a positive-energy one for a minimally coupled, massless scalar field in two-dimensional Minkowski space. We obtain explicit expressions for the maximum time delay and the amount of over-compensation needed, using a previously developed eigenvalue approach. These results are then used to give a proof of the quantum interest conjecture for massless scalar fields in two dimensions, valid for general energy distributions.